L(s) = 1 | − 2.62·2-s + 4.88·4-s − 5-s − 2.62·7-s − 7.56·8-s + 2.62·10-s − 6.72·13-s + 6.88·14-s + 10.0·16-s − 1.78·17-s − 1.48·19-s − 4.88·20-s − 5.20·23-s + 25-s + 17.6·26-s − 12.8·28-s − 1.17·29-s − 4.56·31-s − 11.3·32-s + 4.67·34-s + 2.62·35-s − 0.679·37-s + 3.88·38-s + 7.56·40-s − 5.49·41-s + 3.80·43-s + 13.6·46-s + ⋯ |
L(s) = 1 | − 1.85·2-s + 2.44·4-s − 0.447·5-s − 0.991·7-s − 2.67·8-s + 0.829·10-s − 1.86·13-s + 1.83·14-s + 2.52·16-s − 0.432·17-s − 0.339·19-s − 1.09·20-s − 1.08·23-s + 0.200·25-s + 3.46·26-s − 2.42·28-s − 0.218·29-s − 0.819·31-s − 2.00·32-s + 0.802·34-s + 0.443·35-s − 0.111·37-s + 0.630·38-s + 1.19·40-s − 0.858·41-s + 0.579·43-s + 2.01·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.03132936188\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03132936188\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 7 | \( 1 + 2.62T + 7T^{2} \) |
| 13 | \( 1 + 6.72T + 13T^{2} \) |
| 17 | \( 1 + 1.78T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 + 5.20T + 23T^{2} \) |
| 29 | \( 1 + 1.17T + 29T^{2} \) |
| 31 | \( 1 + 4.56T + 31T^{2} \) |
| 37 | \( 1 + 0.679T + 37T^{2} \) |
| 41 | \( 1 + 5.49T + 41T^{2} \) |
| 43 | \( 1 - 3.80T + 43T^{2} \) |
| 47 | \( 1 + 6.66T + 47T^{2} \) |
| 53 | \( 1 + 14.2T + 53T^{2} \) |
| 59 | \( 1 - 3.88T + 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 - 4.66T + 67T^{2} \) |
| 71 | \( 1 + 3.88T + 71T^{2} \) |
| 73 | \( 1 + 3.56T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 7.33T + 83T^{2} \) |
| 89 | \( 1 + 3.44T + 89T^{2} \) |
| 97 | \( 1 - 1.47T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.168959550531525320691380643747, −7.61142967447168778814151386350, −6.95591562730366245505280626255, −6.50829972009206482264357584046, −5.56004532470682275031298899938, −4.43375737873465653814496821363, −3.29189308787805387899663069070, −2.52385230180130357643559003931, −1.71064908616434207570877179226, −0.11846968641252437367405092367,
0.11846968641252437367405092367, 1.71064908616434207570877179226, 2.52385230180130357643559003931, 3.29189308787805387899663069070, 4.43375737873465653814496821363, 5.56004532470682275031298899938, 6.50829972009206482264357584046, 6.95591562730366245505280626255, 7.61142967447168778814151386350, 8.168959550531525320691380643747